Question

graph the system below and write its solution.\

$$\begin{cases}y = -\\dfrac{1}{2}x + 1 \\\\ 2x + y = 7\\end{cases}$$

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note that you can also answer
o solution\ or \infinitely many\ solutions.\
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solution: \square

Explanation:

Step1: Rewrite second equation in slope-intercept form

$y = -2x + 7$

Step2: Set equations equal to each other

$-\frac{1}{2}x + 1 = -2x + 7$

Step3: Solve for $x$

Add $2x$ to both sides: $\frac{3}{2}x + 1 = 7$
Subtract 1 from both sides: $\frac{3}{2}x = 6$
Multiply by $\frac{2}{3}$: $x = 6 \times \frac{2}{3} = 4$

Step4: Substitute $x=4$ to find $y$

$y = -\frac{1}{2}(4) + 1 = -2 + 1 = -1$
(Verify with second equation: $2(4) + y =7 \implies 8 + y=7 \implies y=-1$)

Answer:

$(4, -1)$

Graphing Notes (for reference):

1. For $y=-\frac{1}{2}x+1$: plot y-intercept $(0,1)$, then use slope $\frac{-1}{2}$ to plot $(2,0), (4,-1)$
2. For $2x+y=7$: plot y-intercept $(0,7)$, then use slope $-2$ to plot $(1,5), (4,-1)$
3. The lines intersect at $(4, -1)$